Following Ref. [1]}, we assume that the quantum vacuum \(\mathinner {|{0}\rangle }\)  corresponds
to a spacetime devoid of any matter or gravitational excitations.
In order to effectively describe the gravitational excitations giving rise to the geometry,
we first rescale the potential \(V_{\rm RN}\)  so as to obtain a canonically normalised real scalar field
\(\Phi =\sqrt{{m_{\rm p}}/{\ell _{\rm p}}}\, V_{\rm RN}\) , and then quantise \(\Phi \)  as a massless field satisfying
the free wave equation in the Minkowski spacetime
\(\left[-\frac{\partial ^2}{\partial t^2}+\frac{1}{r^2}\,\frac{\partial }{\partial r}\left(r^2\,\frac{\partial }{\partial r}\right)\right]\Phi (t,r)=0\ .\) 
